The locus of a point P(,) moving under the condition that the line y= x+ is a tangent to t…

Question

The locus of a point $P(\alpha,\beta)$ moving under the condition that the line $y=\alpha x+\beta$ is a tangent to the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ is:

Accepted Answer

Tangent to the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ is

$$y = mx \pm \sqrt {{a^2}{m^2} - {b^2}} $$

Given that $$y = \alpha x + \beta $$ is the tangent of hyperbola

$$ \Rightarrow m = \alpha $$ and $${a^2}{m^2} - {b^2} = {\beta ^2}$$

$$\therefore$$ $${a^2}{\alpha ^2} - {b^2} = {\beta ^2}$$

Locus is ${a^2}{x^2} - {y^2} = {b^2}$ which is hyperbola.

Solutions · 1